Mathematical Biosciences 201 (2006) 90–100
www.elsevier.com/locate/mbs
Pulse replications and spatially differentiated structure
formation in one-dimensional lattice dynamical system
Akinori Awazu
a,*
, Kunihiko Kaneko
b
a
b
Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Department of Pure and Applied Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan
Received 22 July 2004; received in revised form 14 July 2005; accepted 3 December 2005
Available online 3 February 2006
Abstract
Replication and differentiation of pulses are studied through a simple dynamical system on a one-dimensional lattice. Depending on the parameter values, our proposed system exhibits the following types of
pulse replications: (I) each pulse replicates by itself, (II) each pulse replicates if two or more pulses collide
each other, and (III) pulses replicate by the organization of pulse generators. Moreover, replicated pulses
construct the spatial–temporal pattern when the number of pulses is large. By selecting local objects in such
spatial–temporal patterns and replanting them numerically, it is found that there are two types differentiated local structures: one that continues the production of pulses and the other that does not.
Ó 2005 Elsevier Inc. All rights reserved.
1. Introduction
Recently, simple chemical reaction–diffusion systems have been studied extensively. Some of
these models are known to exhibit the pulse replication [1–9], as has been observed in experiments
[10].
In each model system studied to this time, only a single type of pulses appears, and there is no
differentiation to create different types of pulses or different types of local structures. To search for
*
Corresponding author.
E-mail address: [email protected] (A. Awazu).
0025-5564/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2005.12.013
A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
91
possibility in complex pattern formation as in biological systems, it is interesting to study a simple
reaction–diffusion system displaying not only the replication but also differentiation of local pulse
patterns. We consider a simple dynamical system on a one-dimensional lattice, with one chemical
component. The model shows rather complex dynamics. In particular, it is found that the spatial–
temporal pattern of pulses has two distinct local structures with different dynamical behaviors. We
study how this differentiation of pulse structures appears.
2. Model
As a simple lattice dynamical system [8], we study the following model, possessing only one
component,
u_ i ¼ vi u2i ui þ Dðuiþ1 2ui þ ui1 Þ;
ð1Þ
vi ¼ F ðui ; ui1 ; uiþ1 ; ui2 ; uiþ2 ; ui3 ; uiþ3 ; . . .Þ.
ð2Þ
Here, ui denotes the concentration of the chemical component at the ith site of the lattice. The
diffusion constant is denoted by D.
This model is inspired from Gray–Scott model [1,2], given by,
U_ ¼ VU 2 BU þ DU r2x U;
V_ ¼ VU 2 þ Að1 V Þ þ DV r2 V .
x
ð3Þ
ð4Þ
Here, U and V denote the concentration of the chemical components, DU and DV(>DU) are
diffusion constants for each chemical component, with A and B as constant parameters. Now,
we consider the case that the relaxation of V is fast enough compared to the variation of U.
By neglecting the spatial variation, we obtain V = A/(A + U2) from Eq. (4) by setting V_ ¼ 0. If
we alsoR consider the effect of the diffusion, V is expected to be given by the form
V ðxÞ ¼ dx0 W ðx0 xÞA=ðA þ Uðx0 Þ2 Þ, where W(x) is an unknown weight function that decays
to zero as jxj ! 1. Then, V(x) is approximated by some kinds of spatial average of a decrease
function of U(x). Thus, it is natural to choose F(. . .) so that it gives some spatial average of a
decrease function of ui. As a simple choice, we employ a following form
vi ¼
iþL
X
1
að1 un Þ
2L þ 1 n¼iL
in this paper. Hence, we obtain the following equation represented only by ui
(
)
iþL
X
1
að1 un Þ u2i ui þ Dðuiþ1 2ui þ ui1 Þ;
u_ i ¼
2L þ 1 n¼iL
ð5Þ
ð6Þ
where L gives a range for the spatial averaging of ui. This model is found to show a rich behavior
with the change of parameter a or D, in particular for the case with L = 2.
As the initial condition, we choose mainly the following two conditions: (A) ui = uc is given for
a site i = j and ui = 0 are given for sites i 5 j, (B) ui ¼ uci are given for some sites i = j1, i = j2. . .
located in a small area compared to the system’s space and ui = 0 are given for other sites. Here, uc
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A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
a
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
0.4
Standing Pulse
1
2
0.5
4
5
0.6
D
Moving
Pulse
3
1
0.7
4
1
0.8
Uniform
1 ... Kink produces periodic pattern 4 ... Pulse Produced by Pulse Generator
2 ... Self Replication
5... Self Replication in jamming place
3 ... Replication by collision
(Mesh ~ 0.025)
Fig. 1. Rough phase diagram in the D–a plane. Details of each area are given in the text.
is a random number between ((2L + 1)a ((2L + 1)2a 2 4a(2L + 1)(2D + 1))0.5)/2a and
((2L + 1)a + ((2L + 1)2a2 4a(2L + 1)(2D + 1))0.5)/2a. By the initial condition (A), the states
with only one pulse are prepared in initial, and the states with some pulses are prepared in initial
by (B). We employ the periodic boundary condition for all simulations.1
3. Phase diagram and typical behaviors
The model shows a variety of spatial–temporal pattern with pulse replications. Here, we classify
the types of the behaviors by focusing on the case with L = 2. Some characteristic behaviors such
as annihilating pulses observed in the original Gray–Scott model are not found in this system, but
instead, some types of novel pulse replication are observed that are not observed in the Gray–
Scott model. By a straightforward analysis for the fixed point solutions, spatially uniform solutions with ui 5 0 exist for a P 4, while only the solution with ui = 0 is allowed as the spatially
uniform fixed point solution for a < 4. Now, we focus on the study of dynamics of the model
mostly with the parameter region around a 4.
Fig. 1 shows a rough phase diagram of this system with L = 2 in the D–a plane. ‘Standing
Pulse’ is a state with fixed peaks of ui, ‘Moving Pulse’ is a state with moving peaks of ui without
replication and annihilation, and ‘Uniform’ is a spatially uniform state with ui 5 0.
In other (a,D) region, the replication of pulses of ui is observed. In practice, we found several
types of pulse replicating behaviors as follows.
(1) Gray–Scott like replication. In region 1, the pulse replicates itself when the density of pulse is
low. This state is nothing but that observed in the simulation of Gray–Scott model [1,2,7]. In this
case, the pulse replication ceases and the system becomes stationary forming a spatially periodic
pattern of ui when pulses fill throughout the entire system. Fig. 2 shows a typical spatial–temporal
1
We calculate Eq. (6) by Eular’s scheme with the time step dt = 0.01. The same results can be obtained with
dt = 0.025 or dt = 0.003 are used.
A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
93
ui
6000
Time
5000
4000
3000
2000
1000
0
0
50
100
150
200
250
300
350
400
Space
Fig. 2. Spatial–temporal evolution of ui at region 2 in Fig. 1, where D = 0.5, a = 4.0.
evolution of ui at region 2 in the phase diagram (D = 0.5, a = 4.0). In this case, the pulse replicates
itself not only at a region with low pulse density but also at a region with high pulse density, in
contrast to the Gray–Scott model. Similarly to the case with the region 1, the system becomes stationary when pulses fill throughout the entire system. However, the spatial configuration of pulses
in this case is not regular.
(2) Replication by collision. Fig. 3 shows a typical spatial–temporal evolution of ui at the region
3 in the phase diagram (D = 0.55, a = 4.2). In this case, the pulse cannot replicate itself but can
replicate when two or more pulses collide. (Similar pulse replication is observed recently for a partial differential equations system[4].) As in the region 2, the system becomes stationary, forming an
irregular pulse configuration pattern when pulses fill throughout the entire system.
Now, we discuss initial condition dependence of the final pattern. The pulse replications in the
regions 1, 2 and 3 in the phase diagram are observed independently of the selections of the initial
conditions from (A) or (B). On the other hand in the region 4 and 5 of the phase diagram, the
pulse replication is observed only if the condition (B) is prepared as the initial condition. If we
prepare the condition (A) as the initial condition, only moving pulses without replications are
observed. Later, we only employ the condition (B) as the initial condition.
ui
7000
6000
Time
5000
4000
3000
2000
1000
0
0
50
100
150
200
250
300
350
400
Space
Fig. 3. Spatial–temporal evolution of ui at region 3 in Fig. 1, where D = 0.55, a = 4.2.
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(a)
ui
Time
4000
3000
2000
1000
0
(b)
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
ui
Time
6000
5000
4000
3000
(c)
ui
7000
Time
6000
5000
4000
3000
Space
Fig. 4. Spatial–temporal evolution of ui at region 4 in Fig. 1, where D = 0.7, a = 4.0.
(3) Replication by pulse generator. Fig. 4(a) shows the typical spatial–temporal evolution of ui at
the region 4 in the phase diagram (D = 0.7, a = 4.0). In this case, the following two types of characteristic pulses appear: the oscillating pulses which are the pulses with the temporal oscillations
of ui without spatial moving, and propagating pulses which move spatially in a direction with
keeping their forms. Here, the oscillating pulses are generated around the area where ui 5 0 points
exist initially, and propagating pulses are generated around the oscillating pulses. The propagating pulses are produced one after another by oscillating pulses. The characteristics of these two
types of pulses are different, which will be shown later.
(4) Replication by specific pulses. Fig. 5(a) and (b) show the spatial–temporal evolution of ui and
that of peak points of each pulse, for the region 5 in the phase diagram (D = 0.55, a = 4.0). In
Fig. 5(b), circles are located when a pulse is replicated. As shown in the figure, two types of pulses
appear: one that continues the replication and the other that replicates only a few times. The former pulses are originated from the inner area where ui 5 0 points exist initially.
The appearance of the coexistence of two types of pulses is influenced by the initial condition.
Here, we note that for some pulse patterns, its wavelength is a multiple of lattice sites, while for
some others, the pattern is incommensurate. In Fig. 6(a) and (b), we show two typical initial conditions: (a) the area is occupied mostly by commensurate pulse pattern as shown by arrows, and (b)
the area is occupied by incommensurate patterns. Here, the initial condition in Fig. 6(a) leads to
evolution of ui as in Fig. 6(c), while Fig. 6(b) gives an initial condition for the spatial–temporal pattern as in Fig. 5. As shown in Fig. 5, the pulse replication continues until pulses fill throughout the
A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
95
ui
(a)
Time
1500
1000
500
0
0
50
100
150
0
0
50
100
150
200
250
300
350
400
200
250
300
350
400
(b)
Time
1500
1000
500
Space
Fig. 5. Spatial–temporal evolution of ui at region 5 in Fig. 1, where D = 0.55, a = 4.0. (a) The evolution of each site. (b)
The evolution of tops of each pulse.
(a)
ui
(b)
1
00
(c)
25
ui
0
25
Time
3000
2000
1000
0
0
50
100
150
200
250
300
350
400
Space
Fig. 6. (a) and (b) give typical initial conditions, where (b) gives initial condition of the spatial–temporal evolution of ui
in Fig. 5, and (a) gives a initial condition of the other type of the spatial–temporal evolution of ui in (c), where the
parameters are identical with those of Fig. 5.
entire system from the initial condition with the area mostly filled by incommensurate pulse patterns. On the other hand, as in Fig. 6 (c), pulses only propagate without replication eventually when
started from the initial condition constructed only by commensurate patterns. In Section 5, we will
give a qualitative explanation why incommensurate patterns can produce a large number of pulses.
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A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
4. Characteristics of replicating pulses
Now, we focus on the characteristic differences between pulses which generate new pulses and
those which cannot. Fig. 7 shows two types of typical pulses observed in the region 5 of the phase
diagram. The pulses in the figure propagate with a positive velocity, while only the left pulse replicates. In Fig. 7, the tail of the left pulse oscillates slightly. From the oscillation, a new pulse is
produced. The pulses observed in regions 2 and 3 of the phase diagram have the same property
as this left pulse.
This tail oscillation is caused by the following fact. Here, we focus on the behavior of one pulse.
When the diffusion of ui is slow, compared to the increase of ui, ui at the pulse site and its neighboring sites become large. In such a case, vi of the region around the pulse site decreases because
the local average of ui increases. Then, ui of a site i = j (uj) at the middle of this region decreases
and this pules is divided into two pulses.
If the diffusion is actually slow as in the region 2, the above scenario is valid for the pulse replication. On the other hand, if the diffusion is a little larger as in the case with the regions 3 and 5,
ui diffuses a little farther before the decrease of vj is enough to create a new pulse. In such case, uj
increases again little by little in a little while. Thus, the pulse cannot be divided but some parts of
this pulse oscillate for a long time. In the traveling pulse as observed in region 3 or 5, such oscillating parts shift to the tail of the pulse.
In the region 5, these tail oscillations are usually damped slowly with some iterations of replications. However, these oscillations are sometime amplified, for example by collisions between
pulses, and a new pulse is created. In the region 3, not only the diffusion but also the increase
of ui gets larger. Accordingly, this tail oscillation is sustained. When two or more pulses collide
with each other, this oscillation is amplified to create a new pulse.
5. Differentiations of local patterns
With the increase of pulses, there appear some types of spatial–temporal patterns with regards
to the pulse concentrations. In particular, at the regions 4 and 5 of the phase diagram, these patterns of the pulse configuration consist of different types of local pulse structures. In other words,
Replication
Time
ui
Tail oscillation
Space
No oscillation
Fig. 7. Temporal evolution of pulses with and without tail oscillations. D = 0.55, a = 4.2. This figure is plotted by
taking a moving coordinate with the same velocity as the propagating velocity of pulses to the right-hand side.
A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
97
(a)
0.6
(b)
0.2
0
0.1
<|ui|>
<ui>
1
50
100
150
200
250
300
350
400
F1
F1
0.001
O
1e-05
0
F2
F2
50
100
150
200
250
300
350
400
Space
Fig. 8. (a) Time average of ui, after the total system is filled out by pulses. The same parameter as Fig. 5 are chosen and
(b) shows the time averaged ju_ i j corresponding to (a). O, F1 and F2 indicate the Oscillating, F1-type area, and F2-type
areas, respectively.
there are different attracting patterns. Then, it is interesting to study if temporal evolutions from
each of these local pulse structure may lead to different pulse patterns or not. In order to see the
difference in the attractive to each pattern clearly, we introduce the following ‘replant’ experiment.
First, we choose a local structure from the pattern and ‘replant’ it into the homogeneous pattern
with ui = 0. In other words, we restart the simulation from the initial configuration with a given
local structure and ui = 0 otherwise.
First, we classify local structures in the region 4, using this replanting experiment. There appear
two types of characteristic pulses: the oscillating pulses and propagating pulses (Fig. 4(a)), distinguishable by replanting experiment. When an area containing more than about 20 sites with two
or more oscillating pulses is replanted, the pulse replications as in Fig. 4(b) are observed. On the
other hand, only traveling pulses without replications are observed at all times as in Fig. 4(c),
when the replanted area does not include at least two oscillating pulses. Thus, a region including
two oscillating pulses behaves like a pulse generator studied in [5,6]. Furthermore, the pattern of
ui, that appears when the system is filled by pulses has different characteristics. When an area is
replanted from this final pattern, only the traveling pulses without replications are observed.2
Second, we discuss the differentiation of pulses in region 5, in a little detail. In this region, when
the system is filled out by pulses, there appear three types of areas. To see the difference between
these types, we plotted short time averaged spatial profile of ui and ju_ i j in Fig. 8(a) and (b) after the
total system is filled out by pulses. Here, Fig. 8 are obtained by choosing the same parameter and
the same initial condition as Fig. 5. At the area around the middle of Fig. 8(a) and (b), hju_ i jitime
takes a finite and almost constant value, and ui oscillates periodically in time. We name this area
as an oscillating (O) area. In this area, the propagating pulses fill the space and collide with each
other frequently. The time averaged profile of ui is obtained spatially periodic. The remaining area
is called frozen area, because hju_ i jitime is much smaller than that at the oscillating area, and the
pattern ui is almost frozen. The frozen area is the jam of pulses which is originated by the frequent
2
The behavior explained here as a characteristics of region 4, indeed is observed in the middle part of the region, while
the behavior around the lower boundary of the region is slightly different.
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A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
pulse replications. It is noted that this area tends to appear at the space where a group of pulses as
described in Fig. 6(b) is put in initial because pulses replicate frequently around this region. The
frozen area is further divided into two areas: that with spatially periodic pattern, having higher
peaks of ui than that in the oscillating area (which we call F1 type.), and the transition area between the O and the F1 areas, where the pattern changes spatially as shown in Fig. 8 (which is
termed F2 area.). F1 area is occupied by commensurate pulse patterns. On the other hand, as
in Fig. 8(a), F2 type area lies between two areas with different spatial patterns. It does not have
spatially periodic pattern. The F2 type area contains incommensurate pulse patterns.
Now, we study the characteristic behavior of O, F1, F2 areas, by replanting local structures in
each area. Fig. 9(a)–(c) show spatial–temporal evolutions obtained from the replanting from each
area taken from the pattern of Fig. 8(a). The pulse replication similar to that in Fig. 5(a) appears
in Fig. 9(a), by starting from the initial condition with F2 area. On the other hand, by starting
from the initial condition with O or F1 area, as in Fig. 9(b) and (c), only the pulse propagations
are observed at all times except a few initial replications caused by some disturbance of replanting.
The F2 area consists of incommensurate pulse patterns as in Fig. 6(b). In such incommensurate
patterns, the density of ui can be larger at some part, where vi decreases, leading to the decrease of
Time
(a)
ui
1500
1000
500
0
(b)
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
ui
Time
2000
1500
1000
500
0
(c)
ui
Time
2000
1500
1000
500
0
Space
Fig. 9. Typical spatial–temporal evolution of ui after replanting a local structure F2 (a), F1 (b), and O areas (c). The
same parameters are adopted as in Fig. 5.
A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
99
ui. Hence oscillation of pattern may be produced. Indeed, such incommensurate pulse patterns
have oscillatory instability at the tail, when it is replanted. Although the oscillations at the tail
damps slowly with time as a single pulse by itself, they may be amplified by frequent collisions
between pulses, as shown around the middle of Fig. 9(a). By repetition of the collisions, the replication of pulses continues as in Fig. 5. Note that, in order to keep such frequent collisions, we
need to prepare a large enough area, at least including about 50 sites, with more than 7–8 incommensurate peaks.
To sum up the result of the region 5, only an incommensurate pulse pattern maintains potentiality of replication of pulses. It is interesting to note that two local patterns are distinguishable
with regards to the potentiality of replication of pulses.
At the region without the region 4 or 5, the pulse replications can realize independently of the
initial pulse configurations as mentioned in section 3. Then, we always obtain the same results by
the replant experiments in these cases.
6. Summary and discussion
In this paper, we have studied a simple one-dimensional lattice dynamical system with one variable, that shows replication of pulses as well as differentiation of pulse patterns.
By changing parameter values, several phases are classified, with regards to the propagation
and (self-) replication of pulses. Dynamics of pulse patterns are studied at each phase where
the replication of pulses is possible. We find that distinct spatial–temporal structure of pulses is
formed as the number of pulses increases through the replication. Then, we introduced replant
experiment, to see the initial condition dependence of patterns. This replant experiment is carried
out by choosing a local structures of the original pattern and taking an initial condition of the
local structure around the origin, and 0 otherwise. Here we have found that there are two types
of local structures of pulses: one that keeps replication of pulses and the other that does not. The
origin of this difference in replication is discussed in terms of oscillatory dynamics of pulses.
In considering biology-oriented applications of chemical reaction systems, self-reproduction of
a specific structure is important. In contrast to crystal growth, biological replication is possible
only if specific initial conditions of biochemical systems (for a cell) are selected. Hence, it is interesting to study the selection of initial conditions allowing for self-replication. Although the present model is very simple, it may provide a first-step to study such problem in terms of dynamical
systems.
In the present model, we have chosen the diffusive coupling up to the next-nearest-neighbor
sites, i.e., L = 2. For L = 1 (the case with the nearest neighbor coupling), the pulse replication
is not observed, and only the standing pulse is observed (besides the uniform state). For L > 4,
on the other hand, self-replication of pulses is not observed, and only the growth of pulse region
from the boundaries (i.e., the phase 1 of Fig. 1) is observed, as in the Gray–Scott equation. The
self-replications and the differentiation of pulses, reported here, are observed only if the system
has the specific spatial discreteness as the cases 2 6 L 6 4. Indeed, for L = 3 or 4, the similar phenomena are observed as that reported here for L = 2.
Mathematical analysis of several pulse replicating behaviors as well as an extension of the present model to include more than one chemical components will be of importance in future.
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A. Awazu, K. Kaneko / Mathematical Biosciences 201 (2006) 90–100
Acknowledgments
The author A.A. is grateful to H. Takagi, Y. Hayase and K. Fujimoto for useful discussions.
This work is supported by Grant-in-Aids for Scientific Research from the Ministry of Education,
Science and Culture of Japan (11CE2006).
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